Finite Group Databasebeta

← #13161321

Subgroups — #13161321

Order 1728.

1180 conjugacy classes of subgroups; only the 164 normal subgroups are shown.

normal subgroups only

This group has 1180 conjugacy classes of subgroups — too many to draw. Only its 164 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C32×#1007C211yesmaximal, characteristic, complemented
2#11723988C211yesmaximal, characteristic, complemented
3#11723367C212yesmaximal, complemented
4C3×#63663C211yesmaximal, characteristic, complemented
5#11722043C212yesmaximal, complemented
6#10592944—31nomaximal
7#10591646—34nomaximal
8C2×C33×Q8C2211yescharacteristic
9C2×C4⋊(C3⋊(C3⋊D3))C2211yescharacteristic
10C32×#239—22no—
11C32×D3×Q8—22no—
12C32×#235—22no—
13C32×#235C2212yes—
14#88859—22no—
15C4×C3⋊(C3⋊Dic3)C2211yescharacteristic
16C3×#3748C2212yes—
17#88795—22no—
18#88790—22no—
19C32×2−1+4—31no—
20C3×#1007D314yescomplemented
21#64158—32no—
22#64156—31no—
23#64156—34no—
24#64156—64no—
25#63825—38no—
26#63664—32no—
27#63663D311yescharacteristic, complemented
28C3×#836—34no—
29#63568—38no—
30C22×C3⋊(C3⋊D3)—21no—
31C33×Q8—22no—
32C2×C33×C4D412yes—
33C2×C33×C4C2311yescharacteristic
34C2×C3⋊(C3⋊Dic3)—21no—
35C4⋊(C3⋊(C3⋊D3))—22no—
36C32×#67—22no—
37C32×D12—22no—
38C32×D12D412yes—
39C32×C4×D3—24no—
40C32×C4×D3—22no—
41C32×Dic6—22no—
42C32×Dic6D412yes—
43C3×C3⋊(C3⋊C8)—22no—
44#5245—94no—
45#4753—91no—
46C32×#41—32no—
47C32×#41—62no—
48C32×#41—62no—
49C2×C32×Q8—24no—
50C2×C32×Q8—62no—
51C2×C32×Q8C2×D311yescharacteristic
52C2×C32×Q8C2×D314yes—
53C2×C4⋊(C3⋊D3)—31no—
54C2×C4⋊(C3⋊D3)—34no—
55C2×C4⋊(C3⋊D3)—64no—
56C3×#239—28no—
57C3×D3×Q8—28no—
58C3×#235—28no—
59C3×#235C2×D318yes—
60#3755—62no—
61#3753—62no—
62#3752—616no—
63#3752—62no—
64#3752—68no—
65C4×C3⋊Dic3—31no—
66C4×C3⋊Dic3—34no—
67C4×C3⋊Dic3—64no—
68#3748C2×D312yes—
69C3×#208—38no—
70#3717—68no—
71#3714—68no—
72C22×C33C2×D411yescharacteristic
73C2×C3⋊(C3⋊D3)—42no—
74C2×C3⋊(C3⋊D3)—41no—
75C2×C32×D3—22no—
76C33×C4—22no—
77C33×C4C2×D412yes—
78C3⋊(C3⋊Dic3)—41no—
79C32×Dic3—22no—
80C3×2−1+4—34no—
81#1007C3⋊D311yescharacteristic, complemented
82#949—98no—
83#947—94no—
84#947—91no—
85#947—184no—
86#908—92no—
87C3×#101—91no—
88#837—98no—
89#836—34no—
90#825—92no—
91C22×C3⋊D3—68no—
92C22×C3⋊D3—61no—
93C22×C3⋊D3—64no—
94C32×Q8—22no—
95C32×Q8—28no—
96C32×Q8—32no—
97C32×Q8—48no—
98C32×Q8—62no—
99C32×D4—32no—
100C32×D4—62no—
101C32×D4—62no—
102C2×C32×C4—28no—
103C2×C32×C4—24no—
104C2×C32×C4—64no—
105C2×C32×C4—62no—
106C2×C32×C4D1212yes—
107C2×C32×C4#6718yes—
108C2×C32×C4C22×D314yes—
109C2×C32×C4C22×D311yescharacteristic
110C2×C3⋊Dic3—68no—
111C2×C3⋊Dic3—61no—
112C2×C3⋊Dic3—64no—
113C4⋊(C3⋊D3)—62no—
114C4⋊(C3⋊D3)—68no—
115C4⋊(C3⋊D3)—128no—
116C3×#67—28no—
117C3×D12—28no—
118C3×D12#6718yes—
119C3×C4×D3—216no—
120C3×C4×D3—28no—
121C3×Dic6—28no—
122C3×Dic6#6718yes—
123C3⋊(C3⋊C8)—22no—
124C3×C3⋊C8—68no—
125#454—271noSylow
126C2×C33—21no—
127C2×C33#12011yescharacteristic
128C3⋊(C3⋊D3)—81no—
129C32×D3—42no—
130C3×#41—38no—
131C3×#41—68no—
132C3×#41—68no—
133C2×C3×Q8—24no—
134C2×C3×Q8—68no—
135C2×C3×Q8D3214yes—
136C2×C3×Q8C2×C3⋊D311yescharacteristic
137#239—22no—
138D3×Q8—22no—
139#235—22no—
140#235C2×C3⋊D312yes—
141C2×D12—94no—
142C2×D12—91no—
143C2×D12—184no—
144C3×M16—92no—
145#215—188no—
146#213—188no—
147(C2×D3)⋊C4—1816no—
148(C2×D3)⋊C4—188no—
149(C2×D3)⋊C4—182no—
150C4×Dic3—94no—
151C4×Dic3—91no—
152C4×Dic3—184no—
153#208—38no—
154D24—182no—
155#204—182no—
156C22×C32—24no—
157C22×C32—62no—
158C22×C32C2×D1211yescharacteristic
159C22×C32C2×#6714yes—
160C2×C3⋊D3—128no—
161C2×C3⋊D3—1216no—
162C2×C3⋊D3—128no—
163C2×C3⋊D3—122no—
164C2×C3⋊D3—121no—
165C2×C3⋊D3—124no—
166C2×C3×D3—28no—
167C32×C4—22no—
168C32×C4—28no—
169C32×C4—28no—
170C32×C4—48no—
171C32×C4—62no—
172C32×C4D3×D412yes—
173C32×C4C2×#6718yes—
174C3⋊Dic3—128no—
175C3⋊Dic3—124no—
176C3⋊Dic3—121no—
177C3×Dic3—28no—
1782−1+4—31no—
179AGL(1,ℤ/8)—272no—
180#124—271no—
181#104—272no—
182#101—91no—
183C33#45411yescharacteristic, complemented, Sylow
184C22×D3—188no—
185C22×D3—184no—
186C22×D3—181no—
187C3×Q8—28no—
188C3×Q8—22no—
189C3×Q8—38no—
190C3×Q8—48no—
191C3×Q8—68no—
192C3×D4—38no—
193C3×D4—68no—
194C3×D4—68no—
195C2×C3×C4—28no—
196C2×C3×C4—24no—
197C2×C3×C4—616no—
198C2×C3×C4—68no—
199C2×C3×C4#62618yes—
200C2×C3×C4#63812yes—
201C2×C3×C4C2×D3214yes—
202C2×C3×C4C22×C3⋊D311yescharacteristic
203#67—22no—
204C2×Dic3—188no—
205C2×Dic3—184no—
206C2×Dic3—181no—
207D12—22no—
208D12—188no—
209D12—182no—
210D12—368no—
211D12#63812yes—
212C4×D3—24no—
213C4×D3—22no—
214Dic6—22no—
215Dic6#63812yes—
216C3×C8—182no—
217C3⋊C8—68no—
218C2×C32—28no—
219C2×C32—24no—
220C2×C32—21no—
221C2×C32—24no—
222C2×C32—122no—
223C2×C32#88211yescharacteristic
224C2×C32#95314yes—
225C3⋊D3—248no—
226C3⋊D3—244no—
227C3⋊D3—241no—
228C3×D3—48no—
229#41—32no—
230#41—62no—
231#41—62no—
232C2×Q8—62no—
233C2×Q8D3×C3⋊D311yescharacteristic
234C2×D4—271no—
235SD16—542no—
236D8—542no—
237M16—92no—
238C22⋊C4—542no—
239C42—271no—
240C22×C3—24no—
241C22×C3—68no—
242C22×C3C2×#62614yes—
243C22×C3C2×#63811yescharacteristic
244C2×D3—22no—
245C2×D3—368no—
246C2×D3—3616no—
247C2×D3—368no—
248C2×D3—362no—
249C2×D3—361no—
250C2×D3—364no—
251C3×C4—22no—
252C3×C4—28no—
253C3×C4—28no—
254C3×C4—48no—
255C3×C4—68no—
256C3×C4D3×#6718yes—
257C3×C4C2×#63812yes—
258Dic3—22no—
259Dic3—368no—
260Dic3—364no—
261Dic3—361no—
262C32—24no—
263C32#475311yescharacteristic, complemented
264C32#524514yescomplemented
265C23—541no—
266Q8—22no—
267Q8—32no—
268Q8—62no—
269D4—32no—
270D4—62no—
271D4—62no—
272D4—542no—
273C2×C4—64no—
274C2×C4—62no—
275C2×C4—541no—
276C2×C4#629112yes—
277C2×C4C2×D3×C3⋊D311yescharacteristic
278C8—182no—
279C2×C3—28no—
280C2×C3—24no—
281C2×C3—21no—
282C2×C3—24no—
283C2×C3—128no—
284C2×C3#6397714yes—
285C2×C3#6416211yescharacteristic
286D3—42no—
287D3—728no—
288D3—724no—
289D3—721no—
290C22—62no—
291C22—1082no—
292C22—1081no—
293C22C2×#629111yescharacteristic
294C4—22no—
295C4—62no—
296C4—1081no—
297C4D3×#63812yes—
298C3—24no—
299C3#1059164614yescomplemented
300C3#1059294411yescharacteristic, complemented
301C2—21no—
302C2—122no—
303C2—2161no—
304C2#1172363011yescharacteristic
subgroupG/H
centerC2#11723630
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×D3×C3⋊D3
FittingC2×C33×Q8C22
socleC2×C33#120
hypercenterC2×Q8D3×C3⋊D3
hypocenterC33#454
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×Q8

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C32×#1007C2semidirect
2. #11723988C2semidirect
3. #11723367C2semidirect
4. C3×#63663C2semidirect
5. #11722043C2semidirect
20. C3×#1007D3semidirect
27. #63663D3semidirect
81. #1007C3⋊D3semidirect
183. C33#454semidirect
263. C32#4753semidirect
264. C32#5245semidirect
299. C3#10591646semidirect
300. C3#10592944semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C32×#10072
#117239882
#117233672
C3×#636632
#117220432
#105929443
#105916463